The largest, two digit prime factor of `""^(200)C_(100)` is

Options

A. 61
B. 59
C. 17
D. none of these

Answer

To find the largest two-digit prime factor of \( \binom{200}{100} \), we can follow these steps:

### Step 1: Write the formula for \( \binom{200}{100} \)
The binomial coefficient \( \binom{n}{r} \) is defined as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
So, for our case:
\[ \binom{200}{100} = \frac{200!}{100! \times 100!} \]

### Step 2: Identify the prime factors of \( 100! \)
The prime factorization of \( 100! \) will include all prime numbers less than or equal to 100. The two-digit prime numbers less than 100 are:
\[ 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 \]

### Step 3: Determine the contribution of each prime factor in \( \binom{200}{100} \)
To find the contribution of a prime \( p \) in \( \binom{200}{100} \), we can use the formula:
\[ v_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]
where \( v_p(n!) \) is the exponent of the prime \( p \) in the prime factorization of \( n! \).

Thus, for \( \binom{200}{100} \):
\[ v_p\left(\binom{200}{100}\right) = v_p(200!) - 2 \cdot v_p(100!) \]

### Step 4: Calculate \( v_p(200!) \) and \( v_p(100!) \) for each two-digit prime
We will calculate \( v_p(200!) \) and \( v_p(100!) \) for each two-digit prime \( p \) and check if \( v_p\left(\binom{200}{100}\right) > 0 \).

For example, let’s calculate for \( p = 61 \):
- \( v_{61}(200!) = \left\lfloor \frac{200}{61} \right\rfloor + \left\lfloor \frac{200}{61^2} \right\rfloor = 3 + 0 = 3 \)
- \( v_{61}(100!) = \left\lfloor \frac{100}{61} \right\rfloor + \left\lfloor \frac{100}{61^2} \right\rfloor = 1 + 0 = 1 \)

Now, substituting these into our formula:
\[ v_{61}\left(\binom{200}{100}\right) = 3 - 2 \cdot 1 = 1 \]
Since \( v_{61}\left(\binom{200}{100}\right) > 0 \), 61 is a factor.

### Step 5: Repeat for other two-digit primes
You would repeat this calculation for all two-digit primes until you find the largest one that has a positive contribution.

### Conclusion
After checking all two-digit primes, the largest two-digit prime factor of \( \binom{200}{100} \) is **61**.